Suppose that \(f\in C^{\infty }(M)^G\)  allows us to define a Morse Lie groupoid morphism. To prove that \(f^{\prime }\in C^{\infty }(M^{\prime })^{G^{\prime }}\)  induces another Morse Lie groupoid morphism it suffices to show that \(\phi ^\ast F\)  is a Morse Lie groupoid morphism since \(\phi ^*f=\psi ^*f^{\prime }\) . Indeed, the Lie groupoid morphism \(\phi ^\ast F:(K\rightrightarrows N)\rightarrow (\mathbb {R}\rightrightarrows \mathbb {R})\)  is given by the pair \((F\circ \phi ^1,f\circ \phi ^0)\) . Note that if \(x\in M\)  is a critical point of \(f\circ \phi ^0\) , then \(\phi ^0(x)\)  is a critical point of \(f\)  since \(\phi ^0\)  is a surjective submersion. Thus, from Identity (REF ) we get at \(x\)  that
\(\mathcal {H}_x(f\circ \phi ^0)=d\phi ^0(x)^T\cdot \mathcal {H}_{\phi ^0(x)}(f)\cdot d\phi ^0(x).\) 
From [1]} we know that if \(\phi \)  is a Morita map then \(\overline{d\phi ^0}:\nu (\mathcal {O}_x)\rightarrow \nu (\mathcal {O}^{\prime }_{\phi ^0(x)})\)  is a fiberwise isomorphism. Thus, as \(\mathcal {H}_{\phi ^0(x)}(f)\)  is non-degenerate when restricted to \(\nu _{\phi ^0(x)}(\mathcal {O}^{\prime }_{\phi ^0(x)})\)  we conclude that \(\mathcal {H}_x(f\circ \phi ^0)\)  is non-degenerate when restricted to \(\nu _x(\mathcal {O}_x)\) , as desired.
